#6644819

order	512
nilpotency class	3
isoclinism	isoclinic class of #6632088
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1463 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

(1,2)(3,7)(4,6)(5,8)(9,10)(11,17)(12,20)(13,16)(14,19)(15,22)(18,23)(21,24)(25,26)(27,32)(28,34)(29,35)(30,39)(31,36)(33,37)(38,40);(1,3,4,8)(2,5,6,7)(9,11,13,19)(10,14,16,17)(12,21,22,18)(15,24,20,23)(25,27,31,28)(26,32,36,34)(29,38,30,37)(33,35,40,39);(3,8)(5,7)(25,27,31,28)(26,33,36,40)(29,37,30,38)(32,35,34,39);(2,6)(5,7)(9,12,13,22)(10,15,16,20)(11,18,19,21)(14,23,17,24)(25,28,31,27)(26,34,36,32)(29,37,30,38)(33,39,40,35);(10,16)(14,17)(15,20)(23,24)(25,29,31,30)(26,35,36,39)(27,37,28,38)(32,33,34,40)

Polycyclic presentation:

2^9|b^2=fi,c^2=i,d^2=fhi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,d^b=dfh,e^b=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6552015);